Theorem bj-nuliotaALT 33020

Description: Alternate proof of bj-nuliota 33019. Note that this alternate proof uses the fact that iota x ph evaluates to (/) when there is no x satisfying ph (iotanul 5866). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-nuliotaALT	|- (/) = ( iota x A. y -. y e. x )

Distinct variable group:   x, y

Proof of Theorem bj-nuliotaALT

Step	Hyp	Ref	Expression
1		0ss 3972	. 2 |- (/) C_ ( iota x A. y -. y e. x )
2		iotassuni 5867	. . 3 |- ( iota x A. y -. y e. x ) C_ U. { x | A. y -. y e. x }
3		eq0 3929	. . . . . . 7 |- ( x = (/) <-> A. y -. y e. x )
4	3	bicomi 214	. . . . . 6 |- ( A. y -. y e. x <-> x = (/) )
5	4	abbii 2739	. . . . 5 |- { x | A. y -. y e. x } = { x | x = (/) }
6	5	unieqi 4445	. . . 4 |- U. { x | A. y -. y e. x } = U. { x | x = (/) }
7		df-sn 4178	. . . . . 6 |- { (/) } = { x | x = (/) }
8	7	eqcomi 2631	. . . . 5 |- { x | x = (/) } = { (/) }
9	8	unieqi 4445	. . . 4 |- U. { x | x = (/) } = U. { (/) }
10		0ex 4790	. . . . 5 |- (/) e. _V
11	10	unisn 4451	. . . 4 |- U. { (/) } = (/)
12	6, 9, 11	3eqtri 2648	. . 3 |- U. { x | A. y -. y e. x } = (/)
13	2, 12	sseqtri 3637	. 2 |- ( iota x A. y -. y e. x ) C_ (/)
14	1, 13	eqssi 3619	1 |- (/) = ( iota x A. y -. y e. x )

Syntax hints:   -. wn 3   A.wal 1481   = wceq 1483   {cab 2608   (/)c0 3915   {csn 4177   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)